LMS-CMI Research School

Infomation

Overview

Symplectic manifolds arise as phase spaces in the Hamiltonian formulation of classical dynamics. In this setting, fixed energy levels naturally give rise to contact structures on odd-dimensional manifolds. The global topology of symplectic and contact manifolds has been a central area of study since the groundbreaking work of Gromov, Donaldson, Floer and others in the 1980s and 1990s. Contact topology has steadily increased in importance as a subject in its own right, and has been an essential ingredient in some of the biggest results in low-dimensional topology this century, including Kronheimer-Mrowka's proof of Property P and the deep links exhibited by Ozsváth-Szabó and others between Floer-theoretic invariants such as Heegaard Floer homology and the topology of 3-manifolds.

Several key themes have emerged in three dimensional contact topology. These include: Gromov's h-principle, and the resulting flexible/rigid dichotomy; the importance of open book decompositions of manifolds; fillability questions concerning symplectic manifolds bounded by a given contact manifold; and the intimate relationship between symplectic and contact topology and gauge-theoretic invariants of smooth manifolds.

A key result of Borman-Eliashberg-Murphy in 2014 gave a dramatic advance in our understanding of contact topology in higher dimensions: using an h-principle argument, they extended Eliashberg's classification of flexible contact structures in three dimensions to arbitrary dimension. This result generated a great deal of activity in high-dimensional contact topology, leading to a rapidly growing understanding of which aspects of the theory in three dimensions can be generalised to higher dimensions, and of new phenomena which arise in high dimensions.

This LMS-CMI Research School will give students a comprehensive and accessible introduction to key aspects of contact topology in three dimensions and to the new frontier of high-dimensional contact topology. This is a unique opportunity for students and early-career researchers to get a hands-on guided tour of an exciting and fast-developing area of research from a world-leading team of experts.

Each of the above will give a minicourse of 5 lectures, aimed at graduate students with a fairly broad background. These will be supported by teaching assistants who will lead tutorials/problem sessions. Additional lectures will be given by Patrick Massot, Gordana Matić, and Ivan Smith.

There is a registration fee of £250, or £150 for PhD students. Please note we have funds to cover local expenses including accommodation for PhD students and early career researchers, and to help with travel expenses. We also expect to be able to waive the registration fee for applicants for whom it would be a barrier to participation.

Please note that the registration deadline has been extended to April 11, 2016.

Additional Information

Schedule

All talks take place in 416 Mathematics Building (Monday and Tuesday) and 417 Mathematics Building (rest of the week).

Friday

Titles and Abstracts

Open book decompositions and applications, Vincent Colin

Introduction to the Giroux correspondence; reformulation of Heegaard Floer homology using open books in three dimensions; the equivalence of embedded contact homology and Heegaard Floer homology; Heegaard Floer theory in higher dimensions.

Applications of flexibility in high dimensional contact geometry, Emmy Murphy

Overtwistedness in high dimensions; the plastikstufe, loose Legendrians, and +1-surgeries; open books and overtwistedness, and existence of Weinstein cobordisms; the existence of Liouville cobordisms.